Energy based methods applied in mechanics by using the extended Noether’s formalism

Physical systems are modeled by field equations; these are coupled, partial differential equations in space and time. Field equations are often given by balance equations and constitutive equations, where the former are axiomatically given and the latter are thermodynamically derived. This approach is useful in thermomechanics and electromagnetism, yet challenges arise once we apply it in damage mechanics for generalized continua. For deriving governing equations, an alternative method is based on a variational framework known as the extended Noether’s formalism. Its formal introduction relies on mathematical concepts limiting its use in applied mechanics as a field theory. In this work, we demonstrate the power of extended Noether’s formalism by using tensor algebra and usual continuum mechanics nomenclature. We demonstrate derivation of field equations in damage mechanics for generalized continua, specifically in the case of strain gradient elasticity.


Introduction
In rational continuum mechanics, we axiomatically start with the balance equations. In a very formal way established in [1], the governing equations for thermomechanics and electromagnetism are obtained by starting with the balance equations. For thermomechanics there are balance equations of mass, momenta, and energy. For electromagnetism there are F relation (balance of magnetic flux) and balance of electric charge, they lead to M 's equations. These balance equations are called "universal" since they hold for all materials. For a specific application with known materials, equations for stress, heat flux, (internal or free) energy, electric current, electromagnetic force, charge and current potentials are yet to be defined by constitutive equations in order to close the system of equations. The constitutive equations are needed for establishing the material specific behavior into the system.
Starting with [2,3,4], the thermodynamics of irreversible processes has been used for obtaining the constitutive equations in a formal way. There are several similar methods and extensions to this method, we can list at least four famous approaches: The C -N procedure [5], M 's rational thermodynamics [6], non-equilibrium thermodynamics [7], and extended thermodynamics [8]. Thermodynamical approaches consider bulk properties; for surface phenomena such as crack propagation or surface polarization, there are still difficulties in the formal approach based on thermodynamics. Additional axiomatic balance equations are possible for singular surfaces. When it comes to edge effects, further complications arise and there is simply not one method covering all systems. In short, the method is useful in many applications [9], where we start directly with balance equations. If an application demands a generalization of this method, there are technical challenges.
The formal difficulties aside, definition of internal or free energy is challenging as well. Energy is a directly measurable quantity and a theory based on energy is known as the variational formulation with roots over many centuries long [10,11]. In continuum mechanics, variational formulation has been suggested in different settings, in hydrodynamics [12,13], fluid-structure interactions [14,15], in multiphysics [16,17], in dynamics [18,19], in mechanics with dissipation [20,21], and for discrete structures [22]. Specifically for crack propagation, in quasi-static cases by means of an energy conservation, variational approaches exist as well [23] with technical difficulties in the case of applied surface or volume forces [24]. In classical mechanics, invariance of action is postulated by using a L function. The method is useful, yet the motivation is very formal [25]. L function leads to E -L equations that are the field equations for the bulk. In the case of bulk quantities, classical mechanics [26,27,28] is used for displacement in "regular" systems, i.e. no singularities.
Topological methods by using geometric formulations [29,30] have been used for developing a geometric continuum mechanics [31,32,33,34], which are used in multiphysics as well as multiscale theories [35,36]. Such formulations are used for regular domains, in simpler terms, the underlying mathematical structure needs smooth functions only possible if the continuum body has no voids or cracks between massive particles. A crack may be seen as a singular surface in the sense that the displacement value on both sides of the surface is different. Thus, there is a jump in the displacement field across this "singular" surface or simply defect.
In elasticity, a propagating crack is idealized as material (or configurational) forces are acting on this defect [37]. This model of using material stress has been generalized to obtain an energy-momentum tensor in [38], which is used as a starting point for a possible explanation of the surface related phenomena. This energy-momentum tensor is written by using a space-time continuum, where four coordinates are used for a three-dimensional space and one-dimensional time. This usage has been proven to be adequate in the specific and then general relativity [39], where the space is used as a spatial space. A spatial space is coordinates pointing at locations with or without massive particles occupying the space. We may think of a laboratory frame. This four-dimensional notation has been used in the E energy-momentum tensor, but this time in a material frame where coordinates point to the same massive particle (material point or matter). As the matter may move, a continuum body is deforming, we may think of a co-moving frame for space. In non-relativistic approaches, we skip a distinction between time defined on a laboratory or co-moving frame. By decomposing the energy-momentum tensor into a time and space part, out of the latter, the so-called E stress tensor is obtained and used in connection with crack modeling in different formulations [40,41,42,43,44] and also in applications with numerical examples [45,46,47,48,49,50]. For the history of E stress tensor and its use in variational formulation, we refer to [51]. For technical details in surface phenomena [52] and its relation to fracture mechanics, we refer to [53].
Displacement is the primitive function that we calculate in mechanics. In continuum mechanics, functions are regular without singularities. Crack is indeed causing a jump of the displacement, since the bond between material particles has been lost, and its thickness is at a smaller length-scale such that we consider this singularity as a fictitious surface (without its own mass). The E stress tensor is used in [54] in a systematic study in order to define an entropy production on the singular surface. This excellent idea leads naturally to constitutive equations for the velocity of the singular surface that is tantamount to the crack in physics; in the literature, there is no such consideration for suggesting a relation and its experimental verification. Another novel approach is presented in [55], where M 's rational thermodynamics is used to obtain a balance equation with the necessary constitutive equation for the damage parameter. A differential equation related to the E stress tensor is without doubt a possible modeling tool for crack propagation [56,57].
In brittle elasticity, E stress tensor is well-known and applied in mechanics. When it comes to a generalization, for example higher-order theories or in multiphysics, it is unclear if E stress tensor needs to be altered. The reason is that we circumvent ourselves from discussing its derivation by using N 's formalism in continuum mechanics. We aim in this work for applying N 's formalism by using a continuum mechanics jargon and notation: • First, we discuss about variation of fields, which is the core of the subject and is discussed in an abstract setting. We give examples to well-known transformation rules for clarifying their role in this formulation. For the sake of consistency, we include possibly well known concepts from variational calculus and tensor algebra.
• Second, we explain the extended N 's theorem for fields. We emphasize that we demonstrate (discrete) systems as in dynamics only depending on time and also (continuous) systems depending on space and time. It is of utmost importance to distinguish that many variational formulations consider space as a location in the physical space. Historically, N 's theorem has been applied for such systems mainly in electromagnetism since there the fields propagate with or without matter. But herein, we focus on a continuum body, where space is defined as attached to the matter. Therefore, we may call it a "configurational" space, see [58] for the history of this distinction. We obtain the so-called R -T identity in this configurational space.
• Third, we obtain the usual E -L equations. We provide the usual derivation of N 's current leading to conservation laws, by using E -L equations and R -T identity. These conservation laws are known as balance laws in rational mechanics and they are axiomatically given as the starting point.
• Fourth, we revisit the well-known elastodynamics by using the extended N 's formalism. In this way, it is clarified that the balance of momentum is obtained from the E -L equations, balance of energy is acquired from the N 's current, and the E stress tensor is attained by the R -T identity.
With a clear differing in governing equations and their derivations, a generalization of the theory is one step closer.

Variation of fields
We use standard continuum mechanics notation and understand a summation over all repeated indices called E summation convention. Consider a body composed of massive particles, their image is the continuum domain in the space x i . Coordinates label the particles of the body or in other words we use the particle coordinates analogous as in [59]. We use time, t, and space, (x 1 , x 2 , x 3 ), the numerical values of coordinates do not change as long as they denote the same massive particle. Particle is defined as the smallest observable piece of material consisting of an infinitesimal volume element. This volume element is, however, large enough to neglect any quantum phenomena. As the numerical values of x i do not change for the same particle, the particle rests in this "material" frame. Since the equilibrium is defined by referring the material frame, we call it the reference frame. It may be called the inertial frame with respect to which the dynamics of the system is prescribed. An inertial frame is basically the coordinate system labeling particles, since the definition of the inertial frame is the one where the mass rests 1 and thus does not carry any inertial forces [60].
The particle may deviate from its equilibrium position in this material frame. This deviation from the material frame is fully recoverable. The best example is the elasticity. An elastic deformation is such that the particles change their positions by deviating from the reference frame (equilibrium), after unloading, the particles turn to the same equilibrium position. Reference frame or equivalently the equilibrium position of the continuum body may be altered irreversibly. If the motion of the body is such that the system obtains another equilibrium position upon unloading, called the plasticity, then we need different field equations defining the deviation from the equilibrium and the motion (evolution) of the reference frame.
For a general formulation, we start with the continuous transformations that forms the basis of the H -J equation in the N ian formalism. This variation principle leads to governing equations as introduced formally by [61] and [62]. The variation principle is based on the variational calculus that is the algebra of functionals and their extremal values. A functional is a mapping from a set of values to a single scalar value. Suppose that we define a (tensor of rank 0) function L(t; x i , ∂x i /∂t) depending on an independent variable, t, and on variables depending on this independent variable, x i = x i (t) and ∂x i /∂t = ∂x i ∂t (t). For a domain, Ω, of t ∈ Ω, we calculate the integral called action which becomes a form by rewriting, This so-called first integral in U means that the path of the integral does not matter. According to the theory of invariants the differential form dU is an invariant: Its transformation in x i fails to change its numerical value. We discuss this formal property going back to [63] in more detail in what follows.
The transformation is arbitrary and L is called L an. Often, its study is conducted by using differential forms [64]. We will not make much use of this so called exterior calculus and use the fact that tensors in oblique coordinate systems under (affine) transformations produce same calculus as the invariant theory of (differential) forms [65, §9]. In order to introduce the continuous transformation, we use a function taking the variable x i and transforms along ε, as follows: Suppose that x i = (x, y, z) refer to physical coordinates expressed in a Cartesian system and the transformation is the orthogonal rotation around +z-axis, x = x cos(ε) + y sin(ε) , y = −x sin(ε) + y cos(ε) , z = z , where ε is the rotation angle. Suppose that x i = (t, x, y, z) refer to space-time where the transformation is to a moving system in the direction of x. Between inertial systems, the following is called a G an transformation: where ε is the constant velocity. Suppose now that x i = (x, y, z, ct) is space-time in a M an system and a possible special L transformation reads where ε is called rapidity of the transformation and the speed of light, c, is a universal constant, thus we have used (ct) = ct .
The general framework is used for an arbitrary transformation. Hence, we will not need to define it explicitly and use Eq. (3). We restrict that the transformation be linear. When we expand ζ in T series, around ε = 0, then a first order approximation for the transformation reads where the coefficients to the first power, ξ i , are called the generators of the transformation [66, § 4.1]. As at the limit ε → 0, the approximation becomes accurate, ε is often introduced as a "small" parameter. If we define a tensor of rank one and weight zero, A i , in the space x i , then the variation of the tensor due to the change of space becomes In other words, the variation of A i means Standard tensor calculus rules apply, the transformation is invertible We find the variation of a covariant tensor of rank one [65, § 23] in a straight-forward manner Analogously, for a tensor of rank two and weight zero we obtain

Extended N 's theorem
Suppose that L(t; χ i , ∂χ i /∂t) is defined by an independent variable t ∈ [a, b], by variables χ i = χ i (t) and their derivatives with respect to the independent variable ∂χ i /∂t = ∂χ i ∂t (t). The transformation is defined, t = t + ετ and χ i = χ i + εξ i , with the corresponding generators τ, ξ i . If L is a scalar, its numerical value do not change (invariant) with respect to a particular choice of an infinitesimal transformation with the generators τ , ξ i , as follows: We use a simplified notation L = L(t ; χ i , ∂χ i /∂t ) and rewrite L = L . The (action) functional, I = L dt, has its variation As L = L or in other words, δL = 0, we expect that δI is independent of t, χ i . The trivial way is to choose, δI = 0. As the theory is in first order, i.e. in ε 1 , owing to Eq. (7), we acquire δI → 0 faster than ε → 0. By choosing a small ε, linear in ε is an accurate theory in first order. Therefore, we may have a linear in ε difference, where F is an arbitrary (but given) function. The latter brings an important equation: A more general form for the right-hand side fails to exist, since the transformation is linear in ε. Although not written in this way, the formalism goes back to M and has been used for solving differential equations. This formalism is often called a canonical transformation. 2 We transform the functional in the parameter t that produces a right-hand side, if L is in unit of energy then the induced F is in momentum times position. Momentum has a special meaning; therefore, L did name it "living force." This abstract transformation that produces a source term has been used extensively in [67] to derive the principle of least action. Now, we generalize the procedure by defining (t, x i ) as independent variables and φ k = φ k (t, x i ) as field equations to which L an density (per space-time) depends on, We use a volume element, dΣ = √ g dx dt, physically defining the infinitesimal space-time in a frame with the metric tensor, g µν , and its determinant, g = det(g µν ). This frame may be chosen as laboratory or reference frame. Often in N 's formalism, the choice is done for a laboratory frame (spatial system). Herein, we choose it as the reference frame (material system). In continuum mechanics, a current frame is also introduced as a configuration of massive particles in the physical space, it may be visualized as a deforming continuum body because of the deviation of particles from their reference (mostly defined as their equilibrium) positions. We may find such transformations allowing mapping fields between current and reference frames. In a more general manner, we have two frames and use the transformation properties to switch between them. We define the reference frame in a Cartesian system, i.e. dx ≡ dx 1 dx 2 dx 3 , and thus, the transformed frame is oblique but not curvilinear, dx = ε ijk dx 1 i dx 2 j dx 3 k . By constructing a four-dimensional space for space-time with its metric (for continuous transformation), g µν , where µ, ν ∈ {1, 2, 3, 4}, we obtain 2 In solving partial differential equations canonical transformation has another meaning of bringing the set of equations into a J normal form. Here the same name is used for a different formalism. and therefore dΣ = J dΣ or equivalently dx dt = J dx dt. Moreover, we have by taking the determinant of the first equation, we obtain Since the continuous transformations are written as a set with an identity element and its unique inverse transformations, they form a L group according to the group theory. Of course, we want to have mathematical relations transforming as the coordinate system transforms. Therefore, we use tensors in formulating such relations. A tensor of rank 0 is a scalar. We begin with an assertion that the L density is such a proper scalar, L = L , and observe Equivalently, we may begin with a scalar, I = I , for a given transformation and obtain that the L an is an invariant. By choosing a specific type of generators for the arbitrary transformation, say, L transformation I is called a L scalar. We may give an analogy to continuum mechanics by neglecting shortly the time integral; consider I = I means that the energy is the same in both frames. This fact makes sense, but how do we now that the energy density is also the same in both frames such that L = L is also fulfilled? Indeed, we may want that the energy density is the same for corresponding material particles, but how do we enforce this case? Now by using this analogy, we ask and will answer the question: What if we do have a scalar, I = I, however, not a proper scalar, L = L , by definition, what are the additional conditions to be fulfilled in order to satisfy?
For the formulation for fields, we may incorporate the source term in Eq. (16) (right-hand side) in two steps. First, we build up the procedure for fields with the L an in Eq. (17) for arbitrary transformations in space and time. All the transformation is achieved with different generators but one-parameter ε. The variation of the functional in domain Ω for fields becomes since dΣ = J dΣ. Second, from [68] we know that a transformation produces a non-conservative force. In order to implement this concept, we need a clear distinction between conservative and non-conservative forces. A conservative force is derivable from a proper scalar, S, (from an invariant) whereas a non-conservative force does not have this special property, thus, in the general case we may get a tensor rank 1 from a tensor rank 2, S k i , by using its derivative A transformation from one frame (with prime) onto another frame (without prime), a non-conservative force is generated. This fundamental property may be physically understood as a (virtual) work in case of such a transformation. If we suppose that this transformation is between frames like current and reference, the deviation is physically a virtual displacement: δu i = x i − x i . For this transformation we need to supply an energy, i.e. a virtual work into the system, δA = F i δu i , where this non-conservative force is measurable on the reference frame. The virtual work is virtual since the transformation between the frames is nothing physical. The choice of frame, where the fields are described and equations are evaluated, has nothing to do with the system itself. Thus, the work caused by the transformation is virtual. However, the force is real and is measurable. The direction of transformation from the reference to current gives a minus sign This notion is used in elasticity. But the shown principle is applicable for a transformation between laboratory and reference frame as well. Then a transformation from the fixed laboratory frame (control volume) to the co-moving reference frame reads a positive virtual work. This right-hand side of the latter equation is introduced first time in [69] as δI = ε ∂F µ ∂x µ . In this way it is called a divergence invariance, yet the virtual work is the more general form. We emphasize that this virtual work is given by a non-conservative force such that it is possible to begin with δI = ±δR, where δR is a non-conservative (dissipative) work, also called R dissipation. Indeed, the names work, force, and displacement are in harmony with mechanics, but the relation holds true in multiphysics, as also known from dynamics in discrete systems, they may be called "generalized" force or work-conjugate term. In thermodynamics, one often calls them "thermodynamical" forces and fluxes.
The invariance of the L an leads to the R -T identity [66, § 6.5]. The L an depends on the independent variables (t, x i ) and primitive variables (φ k , ∂φ k ∂t , ∂φ k ∂x i ), which depend on the independent variables. We axiomatically assume that primitive variables exist. We stress that the frame is oblique, hence, we skip distinguishing between covariant and partial space derivatives. The same holds for the time derivative since we measure this in the reference or co-moving frame. The linear transformations read where all of them are along the same parameter ε. Since the invariance property L = L asserts the condition to satisfy, L J − L = ∓δA, we can set the variation along one-parameter ε vanish such as: where we utilize a directed differentiation often called a G derivative. Although we have introduced the virtual work as a physical quantity δA = F i δu i with an analogy in mechanics, in the general case, each primitive variable cause virtual work, δA = F i εϕ i . The linear transformations depend on ε, and thus, L as well as J depend on ε. However, the term L does not, therefore, we obtain By using Eq. (18), we obtain and thus, dL dε ε=0 By using a short notation () • = ∂()/∂t and () ,i = ∂()/∂x i , we calculate and d dε (32) Analogously, we acquire and d dε Now by inserting the latter into Eq. (30), we find the R -T identity: This identity [70,71,72] is general and we examine different scenarios in the following: • First term: When L an does not depend on time, ∂L/∂t = 0, and the right-hand side vanishes, F k = 0, then arbitrary transformations in time are allowed and L is conserved. This property is known as "constant energy." • Second term: In the case of homogeneity-L an is constant in space, x k -arbitrary transformation in space is allowed if right-hand side vanishes. The simple example is a free motion of a rigid body.
• Third term: If L an does not depend on primitive fields, φ k , leading to, ∂L/∂φ k = 0, then no supply or volumetric terms apply. In mechanics, supply is because of gravitational or electromagnetic fields.
• Fourth and fifth terms: Whenever the generator of time transformation depends on time or the generator of space transformation on space, we have a term called energy-momentum tensor if time and space are written together. We discuss these terms in the following.
• The terms ∂L/∂(φ k ) • and ∂L/∂φ k ,i are called the conjugated momenta, however, in case of fields this name may be misleading.
This identity is an extension to the classical N ian approach, we may claim that this identity is one step before obtaining "conservation laws." Especially the fourth term is important to notice, multiplied by a minus, it is often introduced as H ian of the system Herein, we see that the term is motivated by the R -T identity. If we use the latter definition for the H ian, its numerically equal to the canonical H ian in a L an formulation [73]. We emphasize that these concepts of L an and H ian are used for systems with t as the only independent variable. We continue the N ian formalism in the following with time and space as independent variables. First, we combine all of the independent variables together as a set a ν = {t, x 1 , x 2 , x 3 }. We may even think of ν = 0, 1, 2, 3 in order to have a 0 = t and a 1 = x 1 , a 2 = x 2 , a 3 = x 3 , for a simpler analogy. Second, the linear transformation is rewritten, a ν = a ν + εα ν . Third, we write the R -T identity once more for L a ν ; φ k (a ν ), φ k ,µ (a ν ) in this notation, where we use () ,ν = ∂()/∂a ν . Now we define the energy momentum tensor just by rewriting the third term in the latter, This term may lead to confusion since the scalar H ian in the setting with t as the sole independent variable has been turned to a tensor rank 2, which we call energy-momentum tensor. We clearly understand that the L function, L, and its invariance is more fundamental than a theory based on the H function, H = H 0 0 , because it fails to have an invariance property in general. In order to circumvent this confusion, Eq. (39) is called the "energy-momentum" tensor, therefore, we have H ian as its time (scalar) part, H, and E stress tensor as its space part H i j . The term P ν k = ∂L/∂φ k ,µ is called a canonical momenta in the case of rigid body motion, where L incorporates the kinetic energy without deformation energy. We use the conjugate term instead, in elasticity, stress is energetic conjugate of strain (derivative of the primitive variable, which is displacement). In this analogy, the space part of the H ian, H i j , is often introduced by a so-called L transformation. Herein, we realize that the same term is directly generated by the N formalism. The result has been obtained in [59], however, there the condition of functional I being extremal has also been used. The procedure herein is different and less restrictive since the energy-momentum tensor asserts only the invariance of the L an but not its density. The invariance and extremal are separate properties, until now, we have only used invariance.

Conservation laws
We will derive conservation laws in two steps, first, we obtain the so-called E -L equations, second, we use them in the R -T identity in order to obtain the conservation laws that are the balance laws derived by using the formalism herein. We begin with α ν ≡ 0, where this restriction means that we neglect for the moment any shift of the independent variables, i.e. no time and space variation. We stress that not only first two terms vanish in Eq. (36) but also fourth and fifth terms are set to zero, leading to This identity is rewritten in an integral form over a space-time domain Ω and integrated by parts, where the surface integral, dS µ = n µ dS, with a surface normal directed in n µ consists of boundaries of the continuum body and time (initial and end time). We are interested in a differential equation within the domain, in other words, values at boundaries are given so no variation is needed, ϕ k = 0 , ∀a µ ∈ ∂Ω. Thus the last integral vanishes and the well-known E -L equations appear This equation is well-known, for example given as the integral L -'A principle in [74,Definition 7.8.4] for discrete systems. We assert it herein as an additional condition to the R -T identity. It is also called an extremal principle, since it is possible to directly obtain this result from assertion in Eq. (27), where the directional derivative vanishes, i.e. it is an extremal. We realize that for a specific case of no space and time translation, both are identical, which we will discuss further in an application. Often, the right-hand side is neglected in fields, we refer to [75] for a connection of the right-hand side with R dissipation function. We emphasize that we have neglected a reference frame evolution and refer to the Appendix 8 for a simple explanation with an application, if otherwise. As the right-hand side is a non-conservative force, we can derive it from a tensor with one rank higher and obtain For the derivation, we have used an oblique but not curvilinear coordinate system, we refer to [65, § 43], [39] for a generic derivation with C symbols. Now we start with Eq. (39) and obtain by using L = L(a µ ; φ k (a µ ), φ k ,ν (a µ )) and chain rule Now by inserting the latter in the R -T identity in Eq. (38), we acquire By assuming that E -L equations hold, we insert Eq. (43) and obtain By renaming the left-hand side as N 's current, J µ , we obtain the corresponding equation: In the case of vanishing right-hand side, S µ k,µ = ±F k = 0, the latter "balance" equations are called conservation laws, J µ ,µ = 0.

Elastodynamics
In order to demonstrate the meaning of conservation laws, we give an example in elastodynamics. The primitive variables, φ k , are the components of the displacement field, u 1 , u 2 , u 3 , expressed in Cartesian coordinates. We model this reversible process-no reference frame evolution-by using the following L an density: where the stiffness tensor, C ijkl , is given for the corresponding material. Mass density, ρ ref. , is defined on the reference frame, thus, it is constant in time. In the case of linear and isotropic material, the stiffness tensor reads C ijkl = λδ ij δ kl + µδ ik δ jl + µδ il δ jk , where the so-called L parameters are given by engineering constants; Y 's modulus, E, and P 's ratio, ν, as follows: We separate time, µ = 0, and space, µ = {1, 2, 3} = i, for a direct analogy with linear elasticity theory. Hence, the aforementioned L an density in space and time reads with the stored energy density, w in J/m 3 . Albeit not immediately obvious, we use a linear strain measure, E ij = 1/2(u i,j + u j,i ), thus, the stored energy density, w = 1/2E ij C ijkl E kl , is objective and reduces to w = 1/2u i,j C ijkl u k,l effected by the minor symmetries, C ijkl = C jikl = C ijlk . In the case of vanishing viscous effects, F k = 0, we rewrite the conservation laws in Eq. (47), as follows: The whole formulation is in material frame, although for simplicity we ignore the difference between reference and current frame. The energy-momentum tensor, time part is H ian and space part is E stress tensor, reads We use its counterpart in space and time N currents become where we have used H 's law, σ ij = C ijkl E kl . Technically, it is the transpose of P stress, but we demonstrate the methodology for small deformations with a linear strain measure and linear material response. The conservation laws, may be rewritten as follows, as they hold for arbitrary transformations, We stress that the transformations in Eq. (26) may be chosen in such a way that the above equation holds. In this way, it is possible to formulate an inverse problem and search for possible transformations by solving so-called K equations from the latter. A direct problem is to test different transformations and find out the consequence of invariance and extremal of I. The former has brought us the R -T identity and the latter E -L equations. By using both of them, we have obtained N 's currents leading to Eq. (56). If we examine a transformation in displacement field, ϕ k = const, and insert it into the latter, we obtain the balance of linear momentum, Since the transformation is constant in space and time, such a displacement is called a rigid body motion. We may write the result on material frame for a domain B with its closure (smooth boundary) ∂B after applying G -O 's theorem, where traction vectort i is defined on the boundary. Without using C 's tetrahedron argumentation or usual balance equations' argumentation, we obtain the result as a consequence of the transformation rule. The justification is obvious that we search for laws holding under rigid body translations. It is possible to call that a chosen symmetry, ϕ k = const, generates a balance law. This result is not an additional balance law, since we have to satisfy E -L equations, leading to the same governing equation. This consequence is in fact the aforementioned relation that R -T identity reduces to the E -L equations for the case of no space and time variations. More interestingly, now we may easily examine other transformation rules in order to acquire additional governing equations.
Analogously, we may examine a time translation τ = const in order to obtain the balance of energy with the total specific (per mass) energy, e in J/kg, A G an transformation, for example ξ i = const, reads This new balance law reduces to the well-known J-integral with the E stress tensor, for the stationary case. If the integral is taken around a crack tip, the value of this integral is seen as an energy release rate for forming a discontinuity (propagating a crack) [76,77,78]. Herein, its dynamical counterpart is acquired by the N formalism, so its interpretation and use is more obvious. In general, we may skip this balance law and hope that it is fulfilled, but actually, adding an additional restriction is a remedy in numerical accuracy related problems [79]. More different transformations may be examined, for example scaling or rotation (leading to the balance of angular momentum) are studied in [80]. A generalization of this formalism for thermoelasticity is possible as in [81].
We have obtained two governing equations to be fulfilled in an isothermal case. One is the term multiplied by ϕ k and the other is the term multiplied by ξ i .

Generalized continua
Without repeating all the analysis, we now address the case if the L an depends also on the second derivative L a ν ; φ k (a ν ), φ k ,µ (a ν ), φ k ,µγ (a ν ) . As the transformation is still linear, we have extra terms in Eq. (35) appearing for the term multiplied by α µ leading to the R -T identity for generalized continua, Therefore, the energy momentum tensor is renewed With the same integral form as in Eq. (41) and in this round twice-integrating by parts, we obtain E -L equations in generalized continua For a direct analogy, we utilize the same notation as in the previous section with an extension, and rewrite the E -L equations: We repeat the same procedure By using the latter in the R -T identity in Eq. (63), we obtain which is rewritten, as follows: Into the latter, we insert the generalized E -L equations in Eq. (67) and obtain generalized N 's current, We follow the same guidelines and generalize to higher order continua for metamaterials. The primitive variable is again "only" the displacement, φ k = u k , expressedn in Cartesian coordinates. But now the second derivative plays a role as well, so we use the following L an density: where rank 4, 5, 6 tensors, C ijkl , D ijklmn , G ijklm , are given for the corresponding metamaterial. Their measurement seems to be challenging [82,83] , are all defined on the reference frame. We redo the same analysis as before and separate time and space, in order to obtain In the case elasticity, S µ k,µ = 0, we acquire the conservation laws in Eq. (47), as follows: Since we are interested in the terms multiplied by ϕ k and ξ i , only the following terms are sought after and For the generalized elasticity, from the terms multiplied by ϕ k , we acquire In the case of a homogeneous material, where ρ ref. is constant in space, and a centrosymmetric metamaterial, G ijklm = 0, analogous to the previous case, we may use stress σ kj = C kjlm u l,m and double stress m kij = D kijlmn u l,mn , in order to obtain For the generalized mechanics, the energy-momentum tensor becomes we write it in terms of space and time, By using the latter, we obtain the generalized J-integral as follows: (82) We emphasize that inertial terms arise on the surface integral. Such a result is challenging to obtain without a formal structure as presented herein. In the case of the stationary case, for a centro-symmetric material, G ijklm = 0, we obtain 0 = ∂B − n i w + n j u k,i C kjlm u l,m + u k,li D kljmno u m,no dA .
By utilizing σ kj = C kjlm u l,m and m klj = D kljmno u m,no for obtaining tractiont k = σ kj n j and double traction s kl = m klj n j , the generalized J-integral reads 0 = ∂B − n i w + u k,itk + u k,liskl dA .
Hence, we understand that even a stable crack propagation is steered by not only traction but also double traction. The latter term may be proposed with the help of its structure, but the term from the kinetic energy u • k,i ρ ref. d 2 ref. u • k,j n j would be missed easily. It is rather difficult to approximate its role since d ref. is challenging to measure, for a numerical analysis with an experimental comparison for the role of this term, we refer to [95].

Conclusion
We have revisited the extended N 's formalism in continuum mechanics by using tensor algebra and applied directly to elastodynamics. Apart the well-known balance equations, we have observed how the Jintegral is obtained with the E stress tensor, which is of importance in modeling damage mechanics. In this way, we understand that this formalism includes all necessary information for a theory and it is more useful in order to extend the conventional mechanics. Extension may be explained for introducing dissipation as a reason of non-local interaction between particles [96], for its English translation, see [97]. Such effects may also be modeled by generalized continua [98,99,100,101], especially at smaller length-scales, where the continuum length-scale converges to the microstructure [102]. By using a variational method, generalized mechanics is acquired in a straight-forward manner [103]. However, its generalization to damage mechanics has difficulties, since damage mechanics is not acquired directly from the variational formalism. One possible approach is a hemivariational approach [104,105,106] but its extension in multiphysics is challenging.
With this work, we expect to shed some light on this formalism and motivate to develop numerical methods based on a purely variational formulation [107]. In this manner, possible explanations arise for difficult concepts such as contact formulations [108]. By using the additional balance law from the extended N formalism is expected to be beneficial in such a setting for modeling the propagation of discontinuities in primitive variables (crack for the displacement field).

Appendix
We demonstrate in a simplified form how the measure gets a role in the irreversibility. Although the given example below is out of our scope in mechanics, it is beneficial to see this relation. Probably, this application is the only physical example, where the metric evolution is known. In cosmology [109] the universe is expanding with a (known) parameter a =ā(t) everywhere the same, leading to the metric for that expanding universe: g ij =   a 2 0 0 0 a 2 0 0 0 a 2   , g = det(g ij ) = a 6 , √ g = a 3 .
Thus, the infinitesimal volume element reads dV = √ g dx = a 3 dx and is time dependent. The rigid motion of galaxies, for example in one direction, χ, will be calculated. We use the same short notation, χ • = ∂χ/∂t, and build the L an in that time dependent (expanding) metric The latter gives the energy density with the ground state V = ρU χ depending solely on the motion χ, the ground state has different names in the literature: dark energy, vacuum energy, as well as cosmological constant. We plug in the L density into the E -L equations, use the material frame, ρ • = 0, and obtain where we have used the so-called H constant h = a • /a. Obviously, due to the expansion with velocity related to h, there is a damping in this partial differential equation such that the process is irreversible. We use this analogy and understand plasticity in mechanics as an irreversible change of the reference frame (herein the metric). Of course, the situation is far more difficult since additional governing equations need to be solved. In this simple example from cosmology, the H constant is a given constant.